Quotient of sum of exps

Percentage Accurate: 98.7% → 100.0%

Time: 7.7s

Alternatives: 14

Speedup: 2.7×

• could not determine a ground truth

  1. a = 7.651187167493154e+121
  2. b = 170488.4559264945

Specification

Math

\[\begin{array}{l} \\ \frac{e^{a}}{e^{a} + e^{b}} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ (exp a) (+ (exp a) (exp b))))

C

double code(double a, double b) {
	return exp(a) / (exp(a) + exp(b));
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = exp(a) / (exp(a) + exp(b))
end function

Java

public static double code(double a, double b) {
	return Math.exp(a) / (Math.exp(a) + Math.exp(b));
}

Python

def code(a, b):
	return math.exp(a) / (math.exp(a) + math.exp(b))

Julia

function code(a, b)
	return Float64(exp(a) / Float64(exp(a) + exp(b)))
end

MATLAB

function tmp = code(a, b)
	tmp = exp(a) / (exp(a) + exp(b));
end

Wolfram

code[a_, b_] := N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{a}}{e^{a} + e^{b}} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ (exp a) (+ (exp a) (exp b))))

C

double code(double a, double b) {
	return exp(a) / (exp(a) + exp(b));
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = exp(a) / (exp(a) + exp(b))
end function

Java

public static double code(double a, double b) {
	return Math.exp(a) / (Math.exp(a) + Math.exp(b));
}

Python

def code(a, b):
	return math.exp(a) / (math.exp(a) + math.exp(b))

Julia

function code(a, b)
	return Float64(exp(a) / Float64(exp(a) + exp(b)))
end

MATLAB

function tmp = code(a, b)
	tmp = exp(a) / (exp(a) + exp(b));
end

Wolfram

code[a_, b_] := N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + e^{b - a}} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ 1.0 (+ 1.0 (exp (- b a)))))

C

double code(double a, double b) {
	return 1.0 / (1.0 + exp((b - a)));
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 1.0d0 / (1.0d0 + exp((b - a)))
end function

Java

public static double code(double a, double b) {
	return 1.0 / (1.0 + Math.exp((b - a)));
}

Python

def code(a, b):
	return 1.0 / (1.0 + math.exp((b - a)))

Julia

function code(a, b)
	return Float64(1.0 / Float64(1.0 + exp(Float64(b - a))))
end

MATLAB

function tmp = code(a, b)
	tmp = 1.0 / (1.0 + exp((b - a)));
end

Wolfram

code[a_, b_] := N[(1.0 / N[(1.0 + N[Exp[N[(b - a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{e^{a}}{e^{a} + e^{b}}} \]

   2. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

   4. div-inv N/A

      \[\leadsto \frac{1}{\color{blue}{\left(e^{a} + e^{b}\right) \cdot \frac{1}{e^{a}}}} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\left(e^{a} + e^{b}\right) \cdot \frac{1}{e^{a}}}} \]

   6. lift-exp.f64 N/A

      \[\leadsto \frac{1}{\left(e^{a} + e^{b}\right) \cdot \frac{1}{\color{blue}{e^{a}}}} \]

   7. rec-exp N/A

      \[\leadsto \frac{1}{\left(e^{a} + e^{b}\right) \cdot \color{blue}{e^{\mathsf{neg}\left(a\right)}}} \]

   8. lower-exp.f64 N/A

      \[\leadsto \frac{1}{\left(e^{a} + e^{b}\right) \cdot \color{blue}{e^{\mathsf{neg}\left(a\right)}}} \]

   9. lower-neg.f64 99.2

      \[\leadsto \frac{1}{\left(e^{a} + e^{b}\right) \cdot e^{\color{blue}{-a}}} \]

4. Applied rewrites 99.2%

   \[\leadsto \color{blue}{\frac{1}{\left(e^{a} + e^{b}\right) \cdot e^{-a}}} \]

5. Taylor expanded in a around inf

   \[\leadsto \frac{1}{\color{blue}{e^{\mathsf{neg}\left(a\right)} \cdot \left(e^{a} + e^{b}\right)}} \]
6. Step-by-step derivation

   1. distribute-lft-in N/A

      \[\leadsto \frac{1}{\color{blue}{e^{\mathsf{neg}\left(a\right)} \cdot e^{a} + e^{\mathsf{neg}\left(a\right)} \cdot e^{b}}} \]

   2. exp-neg N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}}} \cdot e^{a} + e^{\mathsf{neg}\left(a\right)} \cdot e^{b}} \]

   3. lft-mult-inverse N/A

      \[\leadsto \frac{1}{\color{blue}{1} + e^{\mathsf{neg}\left(a\right)} \cdot e^{b}} \]

   4. lower-+.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{1 + e^{\mathsf{neg}\left(a\right)} \cdot e^{b}}} \]

   5. *-commutative N/A

      \[\leadsto \frac{1}{1 + \color{blue}{e^{b} \cdot e^{\mathsf{neg}\left(a\right)}}} \]

   6. neg-mul-1 N/A

      \[\leadsto \frac{1}{1 + e^{b} \cdot e^{\color{blue}{-1 \cdot a}}} \]

   7. prod-exp N/A

      \[\leadsto \frac{1}{1 + \color{blue}{e^{b + -1 \cdot a}}} \]

   8. lower-exp.f64 N/A

      \[\leadsto \frac{1}{1 + \color{blue}{e^{b + -1 \cdot a}}} \]

   9. neg-mul-1 N/A

      \[\leadsto \frac{1}{1 + e^{b + \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}}} \]

   10. unsub-neg N/A

       \[\leadsto \frac{1}{1 + e^{\color{blue}{b - a}}} \]

   11. lower--.f64 100.0

       \[\leadsto \frac{1}{1 + e^{\color{blue}{b - a}}} \]

7. Applied rewrites 100.0%

   \[\leadsto \frac{1}{\color{blue}{1 + e^{b - a}}} \]
8. Add Preprocessing

Alternative 2: 98.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.9999:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.9999) (/ 1.0 (+ 1.0 (exp (- a)))) (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.9999) {
		tmp = 1.0 / (1.0 + exp(-a));
	} else {
		tmp = 1.0 / (1.0 + exp(b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.9999d0) then
        tmp = 1.0d0 / (1.0d0 + exp(-a))
    else
        tmp = 1.0d0 / (1.0d0 + exp(b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.9999) {
		tmp = 1.0 / (1.0 + Math.exp(-a));
	} else {
		tmp = 1.0 / (1.0 + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.9999:
		tmp = 1.0 / (1.0 + math.exp(-a))
	else:
		tmp = 1.0 / (1.0 + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.9999)
		tmp = Float64(1.0 / Float64(1.0 + exp(Float64(-a))));
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 0.9999)
		tmp = 1.0 / (1.0 + exp(-a));
	else
		tmp = 1.0 / (1.0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.9999], N[(1.0 / N[(1.0 + N[Exp[(-a)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.99990000000000001

   1. Initial program 98.6%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{a}}{e^{a} + e^{b}}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. div-inv N/A

         \[\leadsto \frac{1}{\color{blue}{\left(e^{a} + e^{b}\right) \cdot \frac{1}{e^{a}}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\left(e^{a} + e^{b}\right) \cdot \frac{1}{e^{a}}}} \]

      6. lift-exp.f64 N/A

         \[\leadsto \frac{1}{\left(e^{a} + e^{b}\right) \cdot \frac{1}{\color{blue}{e^{a}}}} \]

      7. rec-exp N/A

         \[\leadsto \frac{1}{\left(e^{a} + e^{b}\right) \cdot \color{blue}{e^{\mathsf{neg}\left(a\right)}}} \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{1}{\left(e^{a} + e^{b}\right) \cdot \color{blue}{e^{\mathsf{neg}\left(a\right)}}} \]

      9. lower-neg.f64 98.6

         \[\leadsto \frac{1}{\left(e^{a} + e^{b}\right) \cdot e^{\color{blue}{-a}}} \]

   4. Applied rewrites 98.6%

      \[\leadsto \color{blue}{\frac{1}{\left(e^{a} + e^{b}\right) \cdot e^{-a}}} \]

   5. Taylor expanded in b around 0

      \[\leadsto \frac{1}{\color{blue}{e^{\mathsf{neg}\left(a\right)} \cdot \left(1 + e^{a}\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{e^{\mathsf{neg}\left(a\right)} \cdot \color{blue}{\left(e^{a} + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{1}{\color{blue}{e^{\mathsf{neg}\left(a\right)} \cdot e^{a} + e^{\mathsf{neg}\left(a\right)} \cdot 1}} \]

      3. exp-neg N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}}} \cdot e^{a} + e^{\mathsf{neg}\left(a\right)} \cdot 1} \]

      4. lft-mult-inverse N/A

         \[\leadsto \frac{1}{\color{blue}{1} + e^{\mathsf{neg}\left(a\right)} \cdot 1} \]

      5. *-rgt-identity N/A

         \[\leadsto \frac{1}{1 + \color{blue}{e^{\mathsf{neg}\left(a\right)}}} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{1 + e^{\mathsf{neg}\left(a\right)}}} \]

      7. neg-mul-1 N/A

         \[\leadsto \frac{1}{1 + e^{\color{blue}{-1 \cdot a}}} \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{1}{1 + \color{blue}{e^{-1 \cdot a}}} \]

      9. neg-mul-1 N/A

         \[\leadsto \frac{1}{1 + e^{\color{blue}{\mathsf{neg}\left(a\right)}}} \]

      10. lower-neg.f64 98.1

          \[\leadsto \frac{1}{1 + e^{\color{blue}{-a}}} \]

   7. Applied rewrites 98.1%

      \[\leadsto \frac{1}{\color{blue}{1 + e^{-a}}} \]

   if 0.99990000000000001 < (exp.f64 a)

   1. Initial program 98.8%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{1 + e^{b}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{1 + e^{b}}} \]

      3. lower-exp.f64 98.6

         \[\leadsto \frac{1}{1 + \color{blue}{e^{b}}} \]

   5. Applied rewrites 98.6%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Developer Target 1: 100.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + e^{b - a}} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ 1.0 (+ 1.0 (exp (- b a)))))

C

double code(double a, double b) {
	return 1.0 / (1.0 + exp((b - a)));
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 1.0d0 / (1.0d0 + exp((b - a)))
end function

Java

public static double code(double a, double b) {
	return 1.0 / (1.0 + Math.exp((b - a)));
}

Python

def code(a, b):
	return 1.0 / (1.0 + math.exp((b - a)))

Julia

function code(a, b)
	return Float64(1.0 / Float64(1.0 + exp(Float64(b - a))))
end

MATLAB

function tmp = code(a, b)
	tmp = 1.0 / (1.0 + exp((b - a)));
end

Wolfram

code[a_, b_] := N[(1.0 / N[(1.0 + N[Exp[N[(b - a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024219 
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64

  :alt
  (! :herbie-platform default (/ 1 (+ 1 (exp (- b a)))))

  (/ (exp a) (+ (exp a) (exp b))))